This work is a continuation of ; it deals with rough boundaries in the simplified context of a Poisson equation. We impose Dirichlet boundary conditions on the periodic microscopic perturbation of a flat edge on one side and natural homogeneous Neumann boundary conditions are applied on the inlet/outlet of the domain. To prevent oscillations on the Neumann-like boundaries, we introduce a microscopic vertical corrector defined in a rough quarter-plane. In  we studied a priori estimates in this setting; here we fully develop very weak estimates à la Nečas  in the weighted Sobolev spaces on an unbounded domain. We obtain optimal estimates which improve those derived in . We validate these results numerically, proving first order results for boundary layer approximation including the vertical correctors and a little less for the averaged wall-law introduced in the literature [13, 18].
"Very Weak Estimates for a Rough Poisson-Dirichlet Problem with Natural Vertical Boundary Conditions." Methods Appl. Anal. 16 (2) 157 - 186, June 2009.